অন্তরীকরণ-১ (Differentiation-1)

অনুশীলনী \(9.B / Q.2\)-এর প্রশ্নসমুহ

\[x\] কে পরিবর্তনশীল ধরে নিচের ফাংশনগুলির অন্তরজ নির্ণয় করঃ
\(Q.2.(i)\) \[x^{-7}\]
উত্তরঃ \[-7x^{-8}\]

\(Q.2.(ii)\) \[\frac{1}{5}x^{5}\]
উত্তরঃ \[x^{4}\]

\(Q.2.(iii)\) \[\frac{2}{3}x^{9}\]
উত্তরঃ \[6x^{8}\]

\(Q.2.(iv)\) \[\frac{6}{x^{4}}\]
উত্তরঃ \[-\frac{24}{x^{5}}\]

\(Q.2.(v)\) \[x^{\frac{1}{3}}\]
উত্তরঃ \[\frac{1}{3\sqrt[3]{x^2}}\]

\(Q.2.(vi)\) \[3x^{-\frac{2}{3}}\]
উত্তরঃ \[-\frac{2}{\sqrt[3]{x^{5}}}\]

\(Q.2.(vii)\) \[\sqrt[3]{x^{2}}\]
উত্তরঃ \[\frac{2}{3\sqrt[3]{x}}\]

\(Q.2.(viii)\) \[\sqrt[3]{4x}\]
উত্তরঃ \[\frac{\sqrt[3]{4}}{3\sqrt[3]{x^2}}\]

\(Q.2.(ix)\) \[ax^2-2bx+c\]
উত্তরঃ \[2(ax-b)\]

\(Q.2.(x)\) \[\frac{x^9+x^3}{x^6}\]
উত্তরঃ \[ 3\left(x^2-\frac{1}{x^4}\right)\]

\(Q.2.(xi)\) \[(2x)^n-b^n\]
উত্তরঃ \[ 2^{n}nx^{n-1}\]
\(Q.2.(xii)\) \[x^{3}(x^{2}-x-2)\]
উত্তরঃ \[ x^2( 5x^{2}-4x-6)\]

\(Q.2.(xiii)\) \[2x^{3}-4x^{\frac{5}{2}}+\frac{7}{2}x^{-\frac{2}{3}}+7\]
উত্তরঃ \[6x^{2}-10x^{\frac{3}{2}}+\frac{7}{3\sqrt[3]{x^{5}}}\]

\(Q.2.(xiv)\) \[(x^{2}-5)^{2}\]
উত্তরঃ \[ 4x(x^{2}-5)\]

\(Q.2.(xv)\) \[\left(x+\frac{1}{x}\right)^{2}\]
উত্তরঃ \[ 2x-\frac{2}{x^{3}}\]

\(Q.2.(xvi)\) \[ \sqrt[3]{x}+\frac{1}{\sqrt[3]{x}}\]
উত্তরঃ \[ \frac{1}{3}\left(\frac{1}{\sqrt[3]{x^{2}}}-\frac{1}{\sqrt[3]{x^{4}}}\right)\]

\(Q.2.(xvii)\) \[\frac{5}{x}+\ln x\]
উত্তরঃ \[ \frac{1}{x}\left(1-\frac{5}{x}\right)\]

\(Q.2.(xviii)\) \[ \sqrt{x^5}\]
উত্তরঃ \[ \frac{5}{2}\sqrt{x^3}\]

\(Q.2.(xix)\) \[\frac{1}{\sqrt{x^{-3}}}\]
উত্তরঃ \[ \frac{3}{2}\sqrt{x}\]

\(Q.2.(xx)\) \[ \frac{x^3-x^7}{\sqrt{x}}\]
উত্তরঃ \[ \frac{1}{2}(5\sqrt{x^{3}}-13\sqrt{x^{11}})\]

\(Q.2.(xxi)\) \[\frac{(x-1)^2}{\sqrt[3]{x}}\]
উত্তরঃ \[\frac{1}{3}\left(5\sqrt[3]{x^{2}}-\frac{4}{\sqrt[3]{x}}-\frac{1}{\sqrt[3]{x^{4}}}\right)\]

\(Q.2.(xxii)\) \[ \frac{x^2-25}{x+5}\]
উত্তরঃ \[1\]
নিচের ফাংশনগুলির \[x\]-এর সাপেক্ষে অন্তরজ নির্ণয় করঃ
\(Q.2.(xxiii)\) \[4\sin x-\cos x\]
উত্তরঃ \[4\cos x+\sin x\]

\(Q.2.(xxiv)\) \[5\cos x+\ln x\]
উত্তরঃ \[-5\sin x+\frac{1}{x}\]

\(Q.2.(xxv)\) \[3e^{x}-5\ln x\]
উত্তরঃ \[3e^{x}-\frac{5}{x}\]

\(Q.2.(xxvi)\) \[x-3\log_ax+5\cos x\]
উত্তরঃ \[1-\frac{3}{x\ln a}-5\sin x\]
\(Q.2.(xxvii)\) \[2x^{a}-5e^{x}+b\sin x\]
উত্তরঃ \[2ax^{a-1}-5e^{x}+b\cos x\]

\(Q.2.(xxviii)\) \[ax^{4}-4\log_ax\]
উত্তরঃ \[4(ax^{3}-\frac{1}{x\ln x})\]

\(Q.2.(xxix)\) \[7\log_ax-5\ln x+4\cos x\]
উত্তরঃ \[\frac{7}{x\ln x}-\frac{5}{x}-4\sin x\]
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